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Do you have "align to baseline grid" turned on in your Paragraph panel? I can't quite repeat the behavior you describe, but that does create a similar behavior. If you want to post your file, I'd be happy to take a look.
Okay, I think I see where the problem lies - though you had me stumped for a moment there!
So I see that you have the text box that contains your table assisting you in aligning your table to the center of your page. (i.e. the text box container is arranged around the outside of the page and the table is centered lef/right using paragraph centering and top/bottom using "align center"command from the control bar. While there's nothing particularly wrong with that method, I think it may be the source of your trouble with the alignment changing unexpectedly.
When looking at your file, I found that removing the "align center" and changing instead to "align top" took away the jumpy behavior of the text box. It no longer shifted when I tried to type. It does however move the whole table to the top of the screen. For the moment let's leave it there. Align to Page: center l to r, center t to b.
Your end result should be an item that is still perfectly centered on the page, but that can be type into without any unpredictable jumping around!
Now, let's select the whole text box that contains the table. Click "Left Justify" in the Paragraphs panel. This will move the table to the top left of your document. Now, fit frame to content. Go to the align panel,
Thanks, that is an easy work around that fixes the problem. It's still weird behavior from InDesign though!